Tensor networks are essential computational tools for efficiently representing high-dimensional data and many-body quantum states. They provide a compact way to describe complex systems, making them valuable in areas such as condensed matter physics and quantum information. Endowing these networks with a Riemannian manifold structure opens a natural path for numerical optimization and analysis of these systems, allowing the use of advanced geometric techniques to manipulate and understand their properties.

A central feature of tensor networks is their gauge freedom, which describes the redundancy in their representation. The characterization of this gauge freedom, often articulated through so-called fundamental theorems, is crucial both for understanding the underlying structure of the networks and for designing efficient numerical algorithms. These theorems allow for the identification of intrinsic network properties that are independent of the particular gauge choice.

This work explores the interaction between the Riemannian manifold structure and the gauge freedom for several families of tensor networks. By using group actions and Riemannian submersions, the researchers have succeeded in establishing a Riemannian fundamental theorem for the studied tensor network families. This advance provides a solid theoretical foundation for the development of new optimization and analysis methods in the field of tensor networks, with potential implications for quantum simulation and machine learning.